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Quasi-commutative polynomial algebras and the power property of \(2\times 2\) quantum matrices. (English) Zbl 0874.17015

The author investigates quadratic algebras of a special type with relations quite similar to ordinary commutativity relations. This approach generalizes different examples of quadratic algebras. In the case of two generators the author unifies the definitions of the algebras \(A_q^{2\mid 0} =K\langle x_1,x_2 \rangle/(x_1x_2-q^{-1} x_2x_1)\) and \(A_J= K\langle x_1,x_2 \rangle/(x_1x_2-x_2x_1-x_1^2)\). This allows one to have a common view of some properties of quantum matrices connected with these two algebras. The main result is that suppose \(V\) is a two-dimensional linear space, \(P,Q\in GL(V)\), \(PQ=QP\) and \(tr(QP)\neq 0\), then for any positive integer \(n\) the following equalities hold in \(E^{P,Q}[V^* \otimes V]\): \[ Z^n(P^{-n} Z^n P^n)^s =(Z^n)^s Q^nZ^n Q^{-n}=\text{DET}^n \cdot I. \] This result is a generalization of [Yu. I. Manin, Topics in Noncommutative Geometry, Princeton University Press (1991; Zbl 0724.17007)] and [B. A. Kupershmidt, J. Phys. A, Math. Gen. 25, L1239–L1244 (1992; Zbl 0769.17009)].
Reviewer: Li Fang (Nanjing)

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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